The relationship between adults’ conceptual understanding of inversion and associativity.
Bibliographic record
Abstract
Children's understanding of the mathematical concepts of inversion and associativity are positively related, as measured by the use of conceptually based shortcut strategies on 3-term inversion problems (i.e., a + b - b, d x e / e) and associativity problems (i.e., a + b - c, d x e / f; Robinson & Dubé, 2009; Robinson & Ninowski, 2003). Individuals who use the inversion shortcut (e.g., 3) are more likely to use the associativity strategy (e.g., 3 x 12 / 4. 12 / 4 = 3, 3 x 3 = 9), which is almost never used by an individual who does not also use the inversion shortcut (Robinson & Dubé, 2009). One possible reason for this relationship is that directing attention to the right-most operation during problem solving may be required to prime the conceptually based shortcut strategies for both problem types. This study investigated the relationship between adults' understanding of inversion and associativity. Adults (N = 42) solved inversion and associativity problems in 1 of 2 conditions. The participants were either presented with the left-most operation and then the whole problem or presented with the right-most operation and then the whole problem. A positive relationship between the use of the conceptually based strategies was found, and it was strikingly similar to the relationship found in childhood. There was evidence that the presentation of the right-most operation first primed the inversion shortcut.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".